Elucidating structure–property relationships of guar gum biomolecules: insights from M -polynomial and QSPR modeling

This study investigates the quantitative structure–property relationship (QSPR) modeling of guar gum biomolecules, focusing on their structural parameters. Guar gum, a polysaccharide with diverse industrial applications, exhibits various properties such as viscosity, solubility, and emulsifying ability, which are influenced by its molecular structure. In this research, M -polynomial and associated topological indices are employed as structural descriptors to represent the molecular structure of guar gum. The M -polynomial and associated topological indices capture important structural features, including size, shape, branching, and connectivity. By correlating these descriptors with experimental data on guar gum properties, predictive models are developed using regression analysis techniques. The analysis revealed a strong correlation between the boiling point and molecular weight and all the considered topological descriptors. The resulting models offer insights into the relationship between guar gum structure and its properties, facilitating the optimization of guar gum production and application in various industries. This study demonstrates the utility of M -polynomial and QSPR modeling in elucidating structure–property relationships of complex biomolecules like guar gum, contributing to the advancement of biomaterial science and industrial applications.


Introduction
Guar gum, an innovative agrochemical, is derived from the endosperm of cluster beans.The species Cyamopsis tetragonoloba, belonging to the Leguminosae family, produces the seeds used in guar gum production, and these seeds are resistant to drought (Prem et al., 2005).The concept of transdomestication was introduced by Hymowitz, although the exact origins of this practice are still a matter of dispute.Further information on the subject can be found in Whistler (1948) and BeMiller (2009).Guar gum has recently piqued the curiosity of several experts for numerous reasons.Guar gum powder serves as a thickener, stabilizer, and health management tool in various industries.It consists of galactomannan polysaccharides and can be derived from different sources, including algae, plants, microorganisms, and animals (Hovgaard and Brondsted, 1996).These polysaccharides are characterized by their stability, non-toxicity, hydrophilicity, and biodegradability.Its unique properties, such as high viscosity, solubility, and stability, make it a versatile ingredient with a wide range of applications in industries like food, pharmaceuticals, cosmetics, and oil drilling.Additionally, guar gum is known for its biocompatibility and non-toxic nature, making it suitable for various biomedical and pharmaceutical applications.By studying guar gum, researchers can explore its potential uses, optimize its properties, and develop innovative products with improved performance and functionality.Overall, the selection of guar gum for research offers opportunities to advance knowledge in various fields and contribute to the development of sustainable and highperforming materials and products.
Chemical graph theory is an interdisciplinary field that links chemistry and mathematics.Graph modeling, originating from early chemical experiments, is a crucial aspect of theoretical chemistry.The subfield of cheminformatics analyzes quantitative structure-activity relationships (QSAR) and quantitative structure-property relationships (QSPR) to predict the biological activity and characteristics of guar gum and its derivatives.Utilizing topological indices and physicochemical substances, it is possible to infer the pharmacological activity of these compounds without conducting experiments.Noteworthy, studies related to topological indices and physicochemical substances include Arockiaraj et al. (2023a); Arockiaraj et al. (2023b);and Arockiaraj et al. (2023c).Recently, algebraic polynomials, such as the Hosoya polynomial (Consonni and Todeschini, 2010a) and M-polynomials (Deutsch and Klavar, 2014), have gained prominence in chemistry for determining distance-based topological indices and degree-based topological indices, respectively.The M-polynomial yields closed forms for various degree-based indices, while the Hosoya polynomial focuses on distance-based indices.The M-polynomial, often denoted as M(x, y), is a polynomial used in the study of chemical graph theory, particularly in the enumeration of certain chemical structures known as molecular graphs.The M-polynomial encodes information about the molecular graph's topology, such as its number of vertices, edges, and other structural properties.It has applications in the enumeration of molecular isomers, the prediction of molecular properties, and the study of chemical reactions.Significant knowledge regarding degree-based graph invariants can be found in the M-polynomial literature, including Munir et al. (2016a); Munir et al. (2016b); Munir et al. (2016c); Munir et al. (2016d);and Ajmal et al. (2017).There is a wealth of knowledge regarding degree-based graph invariants in the M-polynomial.
The M-polynomial of graph Γ is defined as where m st (Γ) is the number of edges ]υ ∈ E(Γ) such that {d ] , d υ } = {s, t}.
Table 1 contains some degree-based TIs and the M-polynomial for the graph Γ:D and The topological index, frequently known as the connectedness index, was introduced in 1947 as a result of Weiner's research (Consonni and Todeschini, 2010b).The earliest and most extensively researched topological index was the Wiener index [for further information, see (Wiener, 1947;Gutman et al., 1986)].One of the earliest topological indices, the Randi c index (Randic, 1975), was first introduced by Milan Randi c in 1975 and is represented by the symbol R − 1 2 (Γ).Its definition is as follows: In 1998, Bollobs and Erds (1998) and Amic et al. (1998) independently proposed the general Randi c index, which has been extensively studied for its numerous mathematical properties (Caporossi et al., 2003;Hu et al., 2005).For a detailed survey, refer to Li et al. (2006).The general Randi c index is defined as and Many papers and books, such as Kier and Hall (1976) and Kier and Hall (1986), have been produced on this topological index.For drug design, the Randi c index was recognized.The first and second Zagreb indices were introduced by Gutman and Trinajsti c.They are denoted as follows: , respectively.The reader is referred to Nikolic et al. (2003); Das and Gutman (2004); Gutman and Das (2004); Vukicevic and Graovac (2004);and Trinajstic et al. (2010) for

TopologicalIndex
DerivationfromM(Γ; x, y) further information on these indices.Among several modifications of Zagreb indices, one is the second modified Zagreb index.According to Milicevic et al. (2004), the second modified Zagreb index for a simple connected graph Γ is defined as The symmetric division index (SDD) is particularly useful in determining the total surface area for polychlorobiphenyls (Gupta et al., 2016) based on the discrete Adriatic indices.The symmetric division index of a connected graph G is defined as The harmonic index is an additional Randi c index variation that is described as There is a relationship between graph eigenvalues and the harmonic index (Favaron et al., 1993).Using MathChem, the elegant structure of extremal graphs is used to generate the inverse sum index (Balaban, 1982), a significant predictor of the octane isomer total surface area.
Furtula et al. ( 2010) is credited for the augmented Zagreb index AZI, which is characterized as Graph invariant AZI has higher prediction power than the atom-bond connectivity index (Furtula et al., 2010) and is a useful predictive measure for analyzing the heat of formation in octanes and heptanes (for more detail, see (Estrada et al., 1998;Das, 2010)).A few well-known degree-based topological indices (which are defined in Eqs 1-7) with M-polynomials (Deutsch and Klavar, 2014) are related in the following Table 1.

Methodology
Molecular graphs and vertex and edge partitions are used to modify the molecular structure of guar gum and its chemical derivatives (Figure 1).Topological indices are derived using M-polynomial.Graphical comparisons of the aforementioned defined indices are made using vertex partition, edge partition, and combinatorial computing.The graphical representation of the outcomes and the comparative study of the findings are performed via 2D plotting in Figure 2 and 3D plotting in Figure 3 are shown by utilizing Mathematica software.In this study, physio-chemical properties of the selected guar gum and its derivatives were obtained from ChemSpider, providing a comprehensive dataset for analysis.Several topological indices were calculated using M-polynomials, extracting key molecular information relevant to biological activities.Subsequently, these indices were utilized in quantitative structure-property relationship (QSPR) analysis, employing SPSS software.The process involves constructing linear, quadratic, and logarithmic models to establish correlations between the calculated topological indices and the properties of selected guar gum and its derivatives.This meticulous methodology aims to uncover patterns and relationships within the molecular structures, contributing to a deeper understanding of the properties of selected guar gum and its derivatives.

Main results and discussions
This section outlines our primary analytical findings and subdivides the material into three sections: guar gum, hydroxypropyl guar, and carboxymethyl guar.The M-polynomials and their associated topological indices are derived for the chemical structures of guar gum, which are useful in the QSPR study.Guar gum is a polysaccharide used in various industries including food, pharmaceuticals, and cosmetics.QSPR studies based on M-polynomials and topological indices can help in understanding how its structural features relate to its properties such as viscosity, solubility, and emulsifying ability.By establishing quantitative relationships between structure and properties, researchers can optimize the production and application of guar gum for various industrial purposes.

Guar gum
This section uses various topological indices to analyze the molecular graph of guar gum. Figure 4 shows the vertex and edge partitioning of guar gum.
Theorem 3.2: Consider a molecular graph Γ for guar gum.Then, Proof.Using the M-polynomial of Γ, we obtain the following equation: Proposition 3.3: Consider a molecular graph Γ for guar gum.Then,  Guar gum molecular graph for n = 3.
Frontiers in Chemistry frontiersin.org Proof: From the edge partitioning of guar gum and using the definition M-polynomial of Γ, we obtain the following equations (Eqs 8-16): S x D y g x, y From Table 1, we obtain the following equation: Proof: From the edge partitioning of guar gum and using the definition M-polynomial of Γ, we obtain the following equations (Eqs 17-20): Molecular graph of CMHPG for n = 3. Frontiers in Chemistry frontiersin.org06 I g x, y   21) From Table 1, we obtain the following equation:

Results of HPG and CMG molecular graphs
When the chemical derivatives of guar gum, such as HPG and CMG, were developed into molecular graphs.The results for these chemical derivatives were combined, as shown below, because the vertex and edge partitions are similar.Figure 5 shows the molecular graphs of HPG and CMG.From the vertices and edges of HPG and CMG, we obtain the following equation: Theorem 3.6: Consider a molecular graph Γ for hydroxypropyl Guar and carboxymethyl guar.Then, Proof: From the edge partitioning of HPG and CMG and using the definition M-polynomial of Γ, we obtain the following equation: Proposition 3.7: Consider a molecular graph Γ for hydroxypropyl guar and carboxymethyl guar.Then, Proof: From the edge partitioning of HPG and CMG and using the definition M-polynomial of Γ, we obtain the following equations (Eqs 21-30):  (d υ , d ] ) Frequency (1,3) 4 (2,2) 1 (2,3) 2 (3,3) 3 TABLE 5 Vertex and edge partitioning of galactose.
(d υ , d ] ) Frequency (1,2) 3 (1,3) 1 (2,3) 6 Frontiers in Chemistry frontiersin.org09 Khalid and Yousaf 10.3389/fchem.2024.1410876From Table 1, we obtain the following equation: Proof: From the edge partitioning of HPG and CMG and using the definition M-polynomial of Γ, we obtain the following equations (Eqs 31-34): (d υ , d ] ) Frequency (1,2) 3 (1,3) 8 (2,2) 1 (2,3) 9 (2,4) 3 (3,3) 9 (3,4) 1  of the topological index H).Linear, quadratic, and logarithmic models and their four important metric values are shown in Tables 12-18.The study focuses on the significance of topological descriptors in predicting the molecular structures of gaur gum and its derivatives, which are crucial in predicting the molecular weight of carbohydrates.Quantitative structure-property relationship (QSPR) methodology is a powerful approach used in the field of drug design, material science, environmental chemistry, cheminformatics, and computational chemistry.QSPR methodology focuses on establishing mathematical relationships between the chemical structure of compounds and their properties, allowing for the prediction of properties based on molecular features.One notable difference is that QSPR specifically targets physical and chemical properties of compounds, while QSAR often focuses on biological activities.Additionally, molecular modeling techniques may involve more complex simulations and calculations to predict molecular behavior.Finally, QSPR methodology offers a systematic and quantitative approach to predict physicochemical properties of chemical compounds based on their molecular structure, distinct from other methodologies such as QSAR, MD simulation, DFT, machine learning models, and hybrid QSAR/QSPR models, each with its own advantages and limitations depending on the specific application and research goals.

Conclusion
This study focuses on the analysis of the polysaccharide guar gum and its chemical variants, namely, HPG, CMG, and CMHPG.Initially, molecular graphs are used to represent these polysaccharides, and vertex and edge partitions are defined.The closed form of the M-polynomial is then computed for these molecular graphs, using various topological indices such as Zagreb indices, Randi c index, inverse Randi c index, H index, SDD index, I index, and AZI index.The molecular structures of the four polysaccharides are compared graphically based on these nine degree-based topological indices.It is important to note that polysaccharides are a type of biopolymer and have diverse applications, particularly in food preservation, the pharmaceutical industry, and petroleum extraction.The findings of this research will be valuable for chemists and pharmaceutical researchers in their respective fields of study.The results of this investigation can have various applications in the field of polymer science and material engineering.By analyzing the topological indices of guar gum, researchers can gain insights into its molecular structure, connectivity, and properties.This information can be used to predict and understand the behavior of guar gum in different environments, such as its solubility, viscosity, and interactions with other molecules.Furthermore, these results can help in the design and optimization of guar gum-based products and formulations.By correlating the topological indices with the performance of guar gum in various applications, researchers can tailor its properties to meet specific requirements in industries such as food, pharmaceuticals, cosmetics, and agriculture.Overall, these results can contribute to a better understanding of its structure-property relationships and facilitate the development of innovative products and technologies in diverse fields.

Future work
The authors will investigate the polysaccharide guar gum and its chemical variants with respect to the generalized reverse degree for future work.

TABLE 1 M
-polynomials are used to derive several degree-based topological indices.

TABLE 2
Properties of aforementioned structures.

TABLE 3
Computation of topological indices.

TABLE 4
Vertex and edge partitioning of arabinose.

TABLE 11
Vertex and edge partitioning of raffinose.